Refined criteria toward boundedness in an attraction-repulsion chemotaxis system with nonlinear productions

We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant, whose formulation can be schematized as...ut = .u -.. . (u.v) +.. . (u.w) in . x (0, Tmax), t vt = .v -.(t, v) + f (u) in . x (0, Tmax), tw = .w -.(t, w) + g(u) in . x (0, Tmax). (.) In this problem, . is a bounded and smooth domain of Rn, for n = 2,.,. > 0, f (u), g(u) reasonably regular functions generalizing, respectively, the prototypes f (u) = auk and g(u) =. ul, for some k, l, a,. > 0 and all u = 0. Moreover,.(t, v) and.(t, w) have specific expressions, t. {0, 1} and ff :=.a -... Once for any sufficiently smooth u(x, 0) = u0(x) = 0, t v(x, 0) = t v0(x) = 0 and tw(x, 0) = tw0(x) = 0, the local well-posedness of problem (.) is ensured, and we establish for the classical solution (u, v, w) defined in . x (0, Tmax) that the life span is indeed Tmax =8 and u, v and w are uniformly bounded in . x (0,8) in the following cases: (I) For.(t, v) = ss v, ss > 0,.(t, w) = dw, d > 0 and t = 0, provided (I.1) k< l; (I.2) k, l. . 0, 2n .; (I.3) k = l and ff < 0, or l = k. . 0, 2n . and ff = 0. (II) For.(t, v) = ss v, ss > 0,.(t, w) = dw, d > 0 and t = 1, whenever (II.1) l, k. . 0, 1n .; (II.2) l. . 1n, 1n + 2 n2+4 . and k. . 0, 1n ., or k. . 1n, 1n + 2 n2+4 . and l. . 0, 1n .; (II.3) l, k. . 1n, 1n + 2 n2+4 .. (III) For.(t, v) = 1 | .| . . f (u) and.(t, w) = 1 | .| . . g(u) and t = 0, under the assumptions k< l or (I.3)). In particular, in this paper we partially improve what derived in Viglialoro [Influence of nonlinear production on the global solvability of an attractionrepulsion chemotaxis system. Math Nachr. 2021;294(12):2441-2454] and solve an open question given in Liu and Li [Finite-time blowup in attractionrepulsion systems with nonlinear signal production. Nonlinear Anal Real World Appl. 2021;61:Paper No. 103305, 21]. Finally, the research is complemented with numerical simulations in bi-dimensional domains.